approximate solutions to the scalar wave equation: the decomposition method

1394 J. Opt. Soc. Am. A/Vol. 15, No. 5 /May 1998 V. Lakshminarayanan and S. Varadharajan

Approximate solutions to the scalar waveequation: the decomposition method

Vasudevan Lakshminarayanan

School of Optometry and Department of Physics and Astronomy, University of Missouri—St. Louis, 8001 NaturalBridge Road, St. Louis, Missouri 63121

Srinivasa Varadharajan

Department of Physics and Astronomy, University of Missouri—St. Louis, 8001 Natural Bridge Road, St. Louis,Missouri 63121

Received July 7, 1997; revised manuscript received December 23, 1997; accepted January 20, 1998

Exact solutions can be obtained for electromagnetic wave propagation in a medium with a simple uniformrefractive-index distribution. For more-complex distributions, approximate or numerical methods have to beutilized. We describe an elegant approximation scheme called the decomposition method for nonlinear differ-ential equations, which was introduced by Adomian [Non-linear Stochastic Systems Theory and Applications toPhysics (Kluwer, Dordrecht, The Netherlands, 1989)]. The method is described and applied to waveguideproblems (planar waveguides with step and parabolic refractive-index profiles), and the results are comparedwith those obtained by JWKB and modified Airy function methods. © 1998 Optical Society of America[S0740-3232(98)03905-2]

OCIS codes: 000.4430, 000.3860, 060.2310, 350.7420.

1. INTRODUCTIONMany problems in physical sciences can be succinctly rep-resented by the equation Fc(x) 5 f(x), where F is someoperator, c is the quantity that is solved for, and f(x) is aforce function. In general, exact solutions cannot alwaysbe found, and we usually have recourse to approximatemethods or numerical solutions. One type of equationthat is often used in physics and engineering is

“

2c~x! 1 G2~x!c~x! 5 0. (1)

In optics, Eq. (1) is a differential equation for the propa-gation of electromagnetic waves in an optical mediumcharacterized by a refractive index n(x). In one dimen-sion, Eq. (1) becomes

d2c

d x2 1 G2~x !c 5 0. (2)

For equations of this type, exact solutions can be found insome cases when n(x) is simple. For other cases, ap-proximation methods can be applied. The typical meth-ods include variational,1 JWKB,2–4 and perturbation5

methods. Ghatak et al.4 used a modified Airyfunction3,4,6–8 (MAF) method to find approximate solu-tions to Eq. (2). An approach based on dynamic program-ming was proposed recently.9 Adomian has proposed atechnique to find approximate solutions to nonlinear sto-chastic differential equations.10–15 The basic idea in thismethodology is to separate the differential operator intoinvertible and noninvertible linear parts and determinis-tic and stochastic nonlinear parts. The solution is writ-ten as a sum of components cn , where one obtains c0 byoperating the inverse of the linear part on the force termand by employing the initial or boundary conditions. The

0740-3232/98/051394-07$15.00 ©

other components are obtained from c0 and from the re-maining parts of the differential operator. In Section 2we describe this technique in detail. In Section 3 we ap-ply the technique to wave propagation in a planar wave-guide with a step index and a parabolic-index distributionand compare the results with those obtained from theJWKB and MAF methods. In that section we also brieflydescribe the JWKB and MAF methods. In Section 4 wediscuss the advantages of the decomposition method overother traditional approximation methods and various pos-sible applications of the decomposition method.

2. DECOMPOSITION METHODConsider a general differential equation of the form

Lc 1 Nc 5 f~x !, (3)

where L denotes the invertible linear part of the differen-tial operator and N is the remaining linear part and thenonlinear part together. L21f(x) is assumed to be mea-surable, where L21 is the inverse differential operator.Equation (3) can be rewritten as

c 5 c~0 ! 1 xc8~0 ! 1 L21@ f~x ! 2 Nc#,

c8~0 ! 5dc

d xUx50

, (4)

and the first two terms can be identified as the solution tothe homogeneous equation Lc 5 0.

In the decomposition method, c is given an iterative so-lution of the form

1998 Optical Society of America

V. Lakshminarayanan and S. Varadharajan Vol. 15, No. 5 /May 1998/J. Opt. Soc. Am. A 1395

c 5 (n50

`

cn~x !, (5)

with c0 being defined as

c0 5 c~0 ! 1 xc8~0 ! 2 L21@ f~x !#. (6)

Substituting Eqs. (5) and (6) into Eq. (4), we get

(n50

`

cn 5 c0 2 L21S N(n50

`

cnD . (7)

Comparing terms in Eq. (7), we get

cn11 5 2L21~Ncn!. (8)

Specifically, we have

c1 5 2L21~Nc0!,

c2 5 2L21~Nc1!,

c3 5 2L21~Nc2!,

and so on. Convergence conditions and other math-ematical proofs can be found elsewhere.10–16

The nonlinear part, N, of the differential operator canalso be given a decomposition representation in terms ofthe An polynomials. The formal construction of thesepolynomials is as follows. Let

g~c! 5 Nc. (9)

We represent g(c) as

g~c! 5 (n50

`

lnAn , (10)

where l is a parameter used to keep track of the terms.An are defined as

An 51n!

dngdlnU

l50

. (11)

Through another function, hn , defined as

hn 51n!

dngdc nU

l50

, (12)

we can write An , for first few values of n, as

A0 5 h0~c0!,

A1 5 h1~c0!c1 ,

A2 5 ~1/2!h2~c0!c12 1 h1~c0!c2 ,

A3 5 ~1/3! !h3~c0!c13 1 h2~c0!c1c2 1 h1~c0!c3 .

(13)

Note that all the values of An are functions of cm , wherem < n. Substituting Eqs. (13) and (6) into Eq. (8) andEq. (8) into Eq. (5), we get an accurate solution of c. Thesolution thus obtained is a series solution similar to well-known power series solutions such as Hermite, Legendre,and Laguerre polynomials. Depending on the precisiondesired, the iteration can be stopped at an appropriatevalue of n, and an approximate solution can be obtained.Once again, we refer the reader to Refs. 10–16 for thederivation of Eqs. (13) and for a deeper understanding ofthe mathematics.

3. PLANAR WAVEGUIDESA. Step-Index WaveguideAs a simple application in optics, we consider the propa-gation of electromagnetic waves along a rectangular (pla-nar) waveguide. First we consider the core refractive in-dex to be a constant (step-index planar waveguide):

n~x ! 5 H n1 uxu < r

n2 uxu . r, (14)

where n1 and n2 are the core and the cladding refractiveindices, respectively, and r is the half-width of the wave-guide. We define the normalized distance X 5 x/r, thewaveguide parameter V 5 krAn1

2 2 n22, and the mode

parameter U 5 krAn12 2 b2, where k is a wave vector

and b is a propagation constant. For guided modes,U 5 jp/2 when U 5 V, the cutoff; and as V → `, U→ ( j 1 1)p/2, where j defines the mode.17 With theseparameters the wave equation can be written as

d2c

dX 2 1 U2c 5 0, (15)

with solutions

c j 5cos~UX !

cos U, j even, (16a)

c j 5sin~UX !

sin U, j odd. (16b)

Equations (16) are valid only for uxu , r. For uxu . r thesolutions are exponentially decaying functions of X (seeRef. 17, Chap. 12).

The decomposition solution can be obtained easily. Wedefine L 5 d2/d X 2. Therefore the solution can be writ-ten as

c 5 c~0 ! 1 Xc8~0 ! 2 L21c, (17)

and we define c0 5 c(0) 1 Xc8(0). Using Eq. (8), wecan obtain the higher-order components. For example,for j 5 2, with n1 5 1.5, n2 5 1.45, r 5 4 mm, andl 5 1.3 mm, we get

c0 5 21,

c1 5 4.9348 X 2,

c2 5 24.0587X4, (18)

etc. The solution, therefore, is

c 5 21 1 4.9348 X 2 2 4.0587X4 1 ..., (19)

whereas the exact solution with U 5 2.8390, is

c 5cos~UX !

cos U. (20)

Figure 1 shows the exact solution, the decompositionapproximation with 10 iterations, and decomposition with20 iterations. It can be seen that the 20-term decompo-sition coincides exactly with the exact solution through-out the extent of the waveguide. We can verify this bycalculating the percentage difference between the exactand the decomposition solutions. Figure 2 shows the per-centage difference between the exact solution and the 10-


and the 20-component decomposition solutions. Whereasthe 10-component approximation is erratic beyond uru/2,the 20-term approximation has zero percentage differencethoughout the width of the waveguide.

B. Parabolic-Index WaveguideAs a slightly more complicated example, consider a planarwaveguide with a parabolic-index profile:

n~x ! 5 n0F1 2 2DS xr D 2G , (21)

where n0 is the core center refractive index, r is the half-width of the waveguide, and 2D 5 sin2 uc , with uc beingthe complement of the critical angle of the material of thecore. We chose this particular profile because it lends it-self naturally to a comparative study of exact, WKB,MAF, and decomposition solutions. The wave equationis

Fig. 1. Exact (solid curve), 10-term decomposition (dotted curve)and 20-term decomposition (dashed curve) solutions to the waveequation in a planar waveguide with step-profile refractive-indexdistribution. The third ( j 5 2) mode is considered here.

Fig. 2. Percentage difference between the exact and the 10-termdecomposition solutions (solid curve) and between the exact andthe 20-term decomposition solutions to Eq. (14). The percentagedifference between the exact and the 20-term decomposition so-lutions is zero, and hence its graph coincides with the x axis.

d2c

d x2 1 @k2n2~x ! 2 b2#c 5 0, (22)

where k is a wave vector and b is a propagation constant.Equation (22) can be rewritten as

d2c

d X 2 1 ~l2 2 X 2!c 5 0, (23)

where we have used the definitions l2 5 U2/V, with U2

5 r2(k2n02 2 b2), V 5 krn0A2D, and X 5 AV(x/r).

Equation (23) has bounded solutions if l2 5 2n 1 1.The solutions depend on n and are given by the nth Her-mite polynomial.18 The exact solutions are

cn~X ! 5 S 1

2nn!ApD 1/2

expS 2X 2

2 DHn~X !, (24)

where Hn(X) are the nth Hermite polynomials. We shallnow illustrate the ways of obtaining an approximate so-lution by using WKB, MAF, and Adomian’s decompositionmethods and compare and contrast their relative merits.

1. JWKB ApproximationThe JWKB approximation method is applicable to all dif-ferential equations that are of the form

d2c

d x2 1 G2~x !c~x ! 5 0 (25)

and for all x that are far away from the turning point x0 ,where x0 is the solution of G2(x) 5 0. [To be more pre-cise, the JWKB approximation is valid when u(1/G)3 (dG/d x)u ! G.] The approximation is not valid nearx0 , where the solution approaches infinity. The generalsolution, assuming that G2 is positive, is, x , x0 ,

cJWKB~x ! 5A

AG~x !sinF E

x

x0

G~x !d xG1

B

AG~x !cosF E

x

x0

G~x !d xG . (26)

A and B are determined from the initial conditions forc (0) and c8(0). For x . x0 , the solution is given by

cJWKB~x ! 5B 2 A

A2h~x !expF E

x0

x

h~x !d xG1

B 1 A

2A2h~x !expF2E

x0

x

h~x !d xG , (27)

where h2(x) 5 2G2(x). We obtain the JWKB approxi-mation by assuming that

c ~x ! 5 F~x !exp@6iu~x !#, (28)

which is similar to c (x) 5 F exp(6iGx) for the case of ahomogeneous refractive-index distribution. c as given byEq. (28) is then substituted back into the differentialequation. The resulting equation is then simplified afterthe condition u(1/G)(dG/d x)u ! G is applied, which yieldsthe expressions for F(x) and u(x)2.


2. Modified Airy Function ApproximationAiry functions are solutions to the differential equation

d2c

d x2 2 xc~x ! 5 0. (29)

The Airy functions, denoted usually Ai(x) and Bi(x), aredefined in terms of two other functions, f(x) and g(x), as

Ai~x ! 5 a1 f ~x ! 2 a2 g~x !, (30)

Bi~x ! 5 A3@a1 f ~x ! 1 a2 g~x !#, (31)

where

f~x ! 5 1 113!

x3 1~1 !~4 !

6!x6 1

~1 !~4 !~7 !

9!x9 1 ...,

(32a)

g~x ! 5 x 124!

x4 1~2 !~5 !

7!x7 1

~2 !~5 !~8 !

10!x10 1 ...,

(32b)

where a1 5 Ai(0) 5 0.35502 ... and a2 5 Ai8(0)5 0.25881 .... Note that Eq. (29) is of the form given byEq. (25), where G2(x) 5 2x. Therefore Airy functions arethe solution of Eq. (25) for G of this form. For all otherforms of G we can adopt a method similar to that used inthe JWKB approximation. We try a solution of the form

c~x ! 5 F~x !Ai@j~x !#, (33)

substitute into Eq. (25), and impose conditions on F(x)and j(x) to obtain

j~x ! 5 2F32 E

x

x0

G~x !d xG 2/3, x , x0 , (34a)

j~x ! 5 F32 E

x0

x

h~x !d xG 2/3, x . x0 , (34b)

where x0 and h are the same as described for the JWKBmethod and

F~x ! 51

Aj8~x !. (35)

We can similarly try a solution c (x) 5 G(x)Bi@j(x)#,with G(x) having the same form as F(x); j(x) is as de-fined above. The general solution, therefore, can be writ-ten as

cMAF~x ! 5C

Aj8~x !Ai@j~x !# 1

D

Aj8~x !Bi@j~x !#.

(36)

C and D can be determined from the initial conditions.For the refractive-index profile, we have

G2~X ! 5 l2 2 X 2, (37a)

h2~X ! 5 X 2 2 l2, (37b)

which result in

EX

X0

G~X !d X 5l2p

42

X

2Al2 2 X 2 2

l2

2sin21S X

lD ,

(38a)

EX0

X

h~X !d X 5X

2AX 2 2 l2 2

l2

2lnuX 1 AX 2 2 l2u

1l2

2lnulu, (38b)

where we have used the fact that X0 5 l. To obtain theJWKB solution, all we need to do is to solve for the con-stants A and B, using the initial conditions. For pur-poses of comparison, we gather these data from the exactsolution. We determine c (0) and c8(0) from the exactsolution and set

cJWKB~0 ! 5 c~0 !, cJWKB8 ~0 ! 5 c8~0 ! (39)

and from Eqs. (39) solve for A and B. For example, forthe third excited mode, i.e., n 5 3, we get

c3~0 ! 5 0, c38~0 ! 5 21

4A48/Ap (40)

from the exact solution; from the JWKB approximationwe get

c3~0 ! 5

A sinS l2p

4 D 1 B cosS l2p

4 DAl

, (41a)

c38~0 ! 5

2Al4 cosS l2p

4 D 1 Bl4 sinS l2p

4 Dl

7/2. (41b)

Using the condition given by Eqs. (39), we can now solvefor A and B and obtain

A 5

A3 cosS l2p

4 DAlAp

, (42a)

B 5 2

A3 sinS l2p

4 DAlAp

. (42b)

With the integrals given by Eqs. (38) and A and B asgiven above, the JWKB approximate solution is known forall X . 0. [For all X , 0 the solution is a reflectionabout the y axis for even n (symmetric modes) and reflec-tion about the y axis coupled with reflection about the xaxis for odd n (antisymmetric modes).] Figure 3 showsthe exact, JWKB, and MAF solutions. It can be seen thatthe MAF solution coincides with the exact solution for allvalues of X and with the JWKB solution for all X notclose to the turning point, which is X 5 A7. Figure 4shows the percentage difference between the exact solu-tion and the MAF and JWKB solutions.

To obtain the MAF solution, all we need to do is to de-termine the constants C and D from the initial conditions[j(X) is known because we have already calculated the in-


tegral through which it is defined, Eqs. (38)]. SinceBi(x) is an exponentially growing function for large posi-tive X, we assume that the MAF solution is of the form

c3~X ! 5C

Aj8~X !Ai@j~X !# (43)

and determine C from the condition on its slope at the ori-gin c8(0):

C 53

2

31/3p

7/12711/12A2

71/3Ai@j~0 !# 2 7~3p!

2/3Ai8@j~0 !#. (44)

Substituting Eq. (44) into Eq. (43) gives us the requiredMAF solution. See Fig. 3 for a comparison of the MAFsolution and the exact solution and Fig. 4 for percentagedifferences. We find that this difference shoots up atX 5 A7 for the JWKB solution, which is expected. Butwe also find that, close to X 5 1.2, the JWKB and theMAF solutions diverge. This point is a zero of the solu-tion. There is a slight mismatch between the zeros of the

Fig. 3. Exact, JWKB, and MAF solutions to the wave equationfor a planar waveguide with a parabolic-index profile. The sec-ond antisymmetric mode (n 5 3) is considered. The solid curveis the exact solution. The MAF solution coincides with the exactsolution. The JWKB solution (dashed curve) is seen to divergeat X 5 A7.

Fig. 4. Percentage difference between the exact solution and theJWKB (solid curve) and the MAF (dotted curve) solutions. TheMAF solution has zero percentage error at all values of X exceptnear the origin and the zero crossing (see text).

exact solution and those of the JWKB and MAF solutions,resulting in the huge percentage differences observed atthis point. Also, the MAF solution differs significantlyfrom the exact solution close to the origin. We shall seethat none of these problems arises in the decompositionmethod.

3. Decomposition MethodNow let us find the decomposition solution to the sameproblem. Let us define L 5 d2/d X 2. Therefore we canwrite

c~X ! 5 c~0 ! 1 Xc8~0 ! 2 L21@G~X !c~X !# (45)

and define

c0 5 c~0 ! 1 Xc8~0 ! 5 214 A48

ApX. (46)

The right-hand side of Eq. (46) was obtained from the ex-act solution. Using Eq. (8), we can obtain the followingcomponents:

c0 5 214 A48

ApX,

c1 5 L21@G~X !c0# 5 214 A48

ApS X 5

202

l2X 3

6 D ,

c2 5 L21@G~X !c1# 5 214 A48

ApS X 9

14402

132520

l2X 7

11

120l4X 5D ,

c3 5 L21@G~X !c2# 5 214 A48

ApS X13

224640

259

1108800l2X11 1

1790720

l4X 9 21

5040l6X 7D

•

•

•

•

The decomposition solution is given by

c 5 c0 1 c1 1 c2 1 c3 1 .... (47)

Since one component is derived from the previous one, asimple computer program can be written, for example,with MAPLE V RELEASE 4 software,19 to generate as manycomponents as we desire. We carried out the computa-tions to 10 terms for n 5 3, and the results are displayedin Fig. 5. We also found that, the smaller the value of n,the fewer the number of components required for us to ar-rive at any desired level of precision. This fact is dis-played in Table 1, where we have listed the number of it-erations required for achieving 1% accuracy for X , 4.Figure 6 displays the relative difference.


4. DISCUSSION AND CONCLUSIONWe have introduced a simple method, Adomian’s decom-position method, for obtaining approximate solutions tothe wave equations. The method requires only a working

Fig. 5. Second antisymmetric exact and 10-term decompositionsolutions to the wave equation in the planar waveguide withparabolic refractive index. The decomposition solution differsfrom the exact solution only for large values of X.

Fig. 6. Percentage difference between the exact and the 10-termdecomposition solutions. The difference is less than 1% for mostvalues of X.

Table 1. Number of Iterations for 1% Accuracy byAdomian’s Decomposition Methoda

Mode Order Number of Iterations

0 131 112 103 104 125 136 167 188 20

a The equation is the wave equation in a planar waveguide with aparabolic-index profile.

knowledge of differentiation and integration. Moreover,the method is applicable to all sorts of differential equa-tions, linear and nonlinear, deterministic and stochastic.As we have explained, the method involves separating theinvertible linear part of the differential equation from theremaining linear and nonlinear parts. The solution isthen written as a sum of components for which one com-ponent is obtained by application of the inverse of the in-vertible linear part to the remainder of the equation. Wehave elaborated this method with application to the wave-guide problems by considering two simple refractive-index profiles and a simple geometry. A similar operatortechnique that makes use of the Pade approximation waspresented recently.20

It is important to bear in mind that the decompositionmethod gives fast convergence, and usually we have tocalculate only the first few components to arrive at a goodapproximation, as can be observed from the examplesthat we have given. For the step-profile planar wave-guide, the number of iterations that needs to be done tokeep the difference below 1% is ;15 for lower-ordermodes, and this number increases with mode number j.For the parabolic-index profile, as we have shown inTable 1, the number of components that needs to be in-cluded in the decomposition approximation for the sameaccuracy is, again, ;15 and again depends on the modeconsidered. For many problems the approach to the ex-act solution is much faster, and we need to consider onlyfive or six components to arrive at less than 1% difference.Also, as X increases, better approximations are obtainedby use of more components.

The decomposition solution does not result in any in-finities, as does the JWKB solution. Nor is it conceptu-ally or mathematically so difficult as the MAF solution.The mathematics involved are simple integration and dif-ferentiation. The only trick is to identify and separatethe invertible linear operator. But this is usually astraightforward act. Moreover, although we have em-phasized the decomposition solution to differential equa-tions, the method is applicable to algebraic equationsalso.14

The method that we have described is useful for deter-mining the eigenfunctions of the wave equation. Themethod can also be used to determine the eigenvalues.We refer the reader to Adomian’s book,14 in which he pre-sents a number of example problems. Also, note thatthis method can be used to determine the Green’s func-tion in those cases in which it is difficult to obtain one bythe regular techniques. The idea is to consider theGreen’s function as a sum of components Gi and to solvefor G0 by using the decomposition method. It is easy toshow that c0 is nothing but the integral of G0 and theforce function g(x). The higher components of c canthen easily be calculated in the usual way. We refer thereader to Ref. 21 for a detailed treatment.

REFERENCES AND NOTES1. For a discussion of variational method see, for example, H.

Goldstein, Classical Mechanics, 2nd ed., Addison-WesleySeries in Physics (Addison-Wesley, Reading, Mass., 1980),Chap. 2. For applications to waveguides see A.


Sharma and P. Bindal, ‘‘Analysis of diffused planar andchannel waveguide,’’ IEEE J. Quantum Electron. 29, 150–153 (1993); ‘‘An accurate variational analysis of single modediffused channel waveguides,’’ Opt. Quantum Electron. 24,1359–1371 (1992).

2. N. Froman and P. O. Froman, JWKB Approximation: Con-tributions to the Theory (North-Holland, Amsterdam, 1965).

3. C. M. Bender and S. A. Orzsag, Advanced MathematicalMethods for Scientists and Engineers (McGraw-Hill, NewYork, 1978).

4. A. K. Ghatak, R. L. Gallawa, and I. C. Goyal, Modified AiryFunction and WKB Solutions to the Wave Equation, Na-tional Institute of Standards and Technology Monogr. 176(U.S. GPO, Washington, D.C., 1991).

5. G. Baym, Lectures on Quantum Mechanics (Benjamin, NewYork, 1969).

6. I. C. Goyal, R. L. Gallawa, and A. K. Ghatak, ‘‘An approxi-mate solution to the wave equation—revisited,’’ J. Electro-magn. Waves Appl. 5, 623–636 (1991).

7. I. C. Goyal, R. L. Gallawa, and A. K. Ghatak, ‘‘Approximatesolutions to the scalar wave equations for opticalwaveguides,’’ Appl. Opt. 29, 2985–2990 (1991).

8. A. K. Ghatak, R. L. Gallawa, and I. C. Goyal, ‘‘Accurate so-lutions to Schrodinger’s equation using modified Airy func-tions,’’ IEEE J. Quantum Electron. 28, 400–403 (1992).

9. M. L. Calvo and V. Lakshminarayanan, ‘‘Light propagationin optical waveguides: a dynamic programming ap-proach,’’ J. Opt. Soc. Am. A 14, 872–881 (1997).

10. G. Adomian, ‘‘Convergent series solution of non-linearequations,’’ J. Comput. Appl. Math. 11, 225–230 (1984).

11. G. Adomian, ‘‘A new approach to the heat equation—an ap-

plication of the decomposition method,’’ J. Math. Anal.Appl. 113, 202–209 (1986).

12. G. Adomian, ‘‘An investigation of asymptotic decompositionmethod for non-linear equations in physics,’’ Appl. Math.Comput. 24, 1–17 (1987).

13. G. Adomian, ‘‘Non-linear oscillations in physical systems,’’Math Comput. Sim. 29, 275–284 (1987).

14. G. Adomian, Non-linear Stochastic Systems Theory and Ap-plications to Physics (Kluwer, Dordrecht, The Netherlands,1989).

15. G. Adomian, ‘‘Non-linear stochastic differential equations,’’in Selected Topics in Mathematical Physics, R. Sridhar, K.Srinivasa Rao, and V. Lakshminarayanan, eds. (Allied,New Delhi, 1995), pp. 47–57.

16. R. Rach, ‘‘On the Adomian decomposition method and com-parisons with Picard’s method,’’ J. Math. Anal. Appl. 128,480–483 (1987).

17. A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall, New York, 1983).

18. M. Abramowitz and I. A. Stegun, Handbook of Mathemati-cal Functions with Formulas, Graphs and MathematicalTables (Dover, New York, 1972).

19. All the computations were done with the commercial soft-ware MAPLE V RELEASE 4, is a product of Waterloo Maple,Inc., Waterloo, Ontario, Canada.

20. C. Yu and D. Yevick, ‘‘Application of the bidirectional para-bolic equations to optical waveguide facets,’’ J. Opt. Soc.Am. A 14, 1448–1450 (1997).

21. I. R. Bellman and G. Adomian, Partial Differential Equa-tions: New Methods for Their Treatment and Solution(Reidel, Dordrecht, The Netherlands, 1985).

approximate solutions to the scalar wave equation: the decomposition method

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